It has been believed that in mathematics, for different NH matrices \(k\) -th
order EPs have the same properties and can be represented by a matrix of
Jordan normal form[1]}, [2]}. In NH physics, the issue of EPs is
related to a NH model that is described by a given Hamiltonian in
corresponding Hilbert space. To emphasize the difference with EPs in
mathematics (the issue about NH matrix), we call EPs in NH physical systems to
be physical EPs. For physical EPs of protected sub-systems in NH
models, we can generalize the definition of EPs by not requiring the
eigenvalues degeneracy (see below discussion). Now, the identification of a
physical EP comes from the coalescence of different eigenvectors. In this
paper, from this starting point, we provide a systematic classification on
physical EPs of protected sub-systems in NH models. In particular, we found
that there may exist a new type of EPs (Basis-type of EPs), for which the
eigenvectors become coalesce without energy degeneracy.
